Classical Electrodynamics · Problem 13.12

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · transition-radiation-two-media

Problem

A relativistic particle of charge zeze moves along the zz axis with a constant speed βc\beta c. The half-space z≤0z \leq 0 is filled with a uniform isotropic dielectric medium with plasma frequency ω1\omega_1, and the space z>0z > 0 with a similar medium whose plasma frequency is ω2\omega_2. Discuss the emission of transition radiation as the particle traverses the interface, using the approximation of Section 13.7. (a) Show that the radiation intensity per unit circular frequency interval and per unit solid angle is given approximately by d2Idω dΩ≈z2e2θ2π2c∣11γ2+ω12ω2+θ2−11γ2+ω22ω2+θ2∣2\frac{d^2I}{d\omega\, d\Omega} \approx \frac{z^2e^2\theta^2}{\pi^2 c} \left|\frac{1}{\dfrac{1}{\gamma^2} + \dfrac{\omega_1^2}{\omega^2} + \theta^2} - \frac{1}{\dfrac{1}{\gamma^2} + \dfrac{\omega_2^2}{\omega^2} + \theta^2}\right|^2 where θ\theta is the angle of emission relative to the velocity of the particle and γ=(1−β2)−1/2\gamma = (1-\beta^2)^{-1/2}. (b) Show that the total energy radiated is I≈z2e23c⋅(ω1−ω2)2(ω1+ω2)⋅γI \approx \frac{z^2e^2}{3c} \cdot \frac{(\omega_1 - \omega_2)^2}{(\omega_1 + \omega_2)} \cdot \gamma

Related theory

Browse the problem library